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aufgerufen, sich an der Entwicklung zu beteiligen! \subsection{Zusammenhangskomponente}\label{graph:komponente} Ein maximaler, zusammenhängender Teilgraph $G^*$ eines Graphen $G$. + \subsection{Wurzelbaum und Suchtheorie} + \subsubsection{Wurzelbaum} + Ein \idx{Wurzelbaum} ist ein Paar $(T,v)$, wobei $T$ ein Baum ist und + $v$ eine ausgezeichnete Ecke, als so genannte \idx{Wurzel}.\\ + Die \idx{Länge einer Ecke} $l(e), e \in E$ bezeichnet den eindeutigen Weg + von $v$ nach $e$. + Die Länge des gesamten Baumes ist die maximale Länge einer Ecke. + + \subsubsection{(n, q)-Baum} + Besitzt ein Baum $n$ Blätter und hat die Wurzel genau $q$ direkte + Nachfolger, spricht man von einem \idx{(n,q)-Baum}. + Besitzt die Wurzel und jede innere Ecke genau $q$ direkte Nachfolger, + ist dies ein \idx{vollständiger (n, q)-Baum} + + \newtheorem{satz}{Satz}[section] + \begin{satz} + Sei $T$ ein (n, q)-Baum, wobei $n \ge 1, q \ge 2$. + Dann ist $l(T) = \log_q n$ + \end{satz} + + \subsubsection{Informationstheoretische Schranke} + Die Menge $\Tau(n,q)$ beschreibt alle (n, q)-Bäume. + Die Zahl $\log_q n = \min\{l(T)|T\in \Tau(n,q)\}$ heisst die + \idx{informationstheoretische Schranke} bzgl. 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z-ca%DxAn(k72d)-DuEumIst3&1-`|J7=e>i!v5O*k$6lcGSx;VV?9RU6r76FFdC=h z42;2<7>lzo4rk*WoQv@)(Z#moJZ!-ExBwU8Qe1?KF@HfNCd%d}sw8`@mtZcg#Lc(` zb1(ywa6N9oEZm5h=*DH3hiRCODYyz(<8n;JHMkbnVLoQ#CR~BZxD^j!A#TG0^x$^f zgGIOlcj7KA#@$$g2eB0Q;yx_Ha`fVUJb)*#0*~Smti@_<#5z2Vr&Qc2_AXAVWOX^; zX2?>CCWxtn%TP zb&tx&LhCM-&uP}reBEvRMCI!b>z69u{y(?pzBWHwyzf-%suuYNhNWx63T19&b98cL jVQmU!Ze(v_Y6^37VRCeMa%E-;Ha0OfHwq;sMNdWwdf|Rx diff --git a/DiMa/dima.tex b/DiMa/dima.tex index 1307f31..c01cf40 100644 --- a/DiMa/dima.tex +++ b/DiMa/dima.tex @@ -93,6 +93,36 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \textbf{Reihenfolge egal, mit Zurücklegen} & \nameref{kombi:mitWiederholung}\\ \end{tabular} +\section{Rekursion} + \subsection{Lineare Rekursion} + Angenommen $T : \N_0 \rightarrow \R $ erfüllt die \idx{lineare Rekursion} + \begin{align*} + T(n) &= r \cdot T(n-1) + a, \; a,r \in \R, n \in \N\\ + T(1) &= b. + \end{align*} + Dann ist die Lösung: + \begin{align*} + T(n) &= r^n T(0) + a \sum_{i=0}^{a-1} r^2, \; n \in \N_0, r = 1 \\ + T(n) &= r^a T(0) + a \frac{1-r^n}{1-r}, \text{\ für } r \ne 1 + \end{align*} + + \subsection{Wachstum von Funktionen} + Seien $f,g : \N_0 \rightarrow \R$.\\ + \idx{Abschätzung nach oben}\\ + Eine Funktion kann nach oben asymptotisch abgeschätzt werden, wenn + $\exists c > 0$, s.d. $|f(n)| \le c \cdot |g(n)|, \text{\ für } n \ge n_0$. + Man schreibt dann $f(n) = O(g(n))$, $O(g(n))$ enthält also alle Funktionen, + durch die $f(n)$ nach oben asymptotisch abgeschätzt wird. + + \idx{Abschätzung nach unten}\\ + Wenn $\exists c > 0$, so dass $|f(n)| \ge c \cdot |g(n)|$, schreibt man + $f(n) = \Omega(g(n))$. So wächst $f(n)$ also schneller als $g(n)$. + + \idx{Abschätzung nach oben und unten} + Die Kombination von $O$ und $\Omega$ heißt $\Theta$. + Also ist $f(n) = \Theta$, wenn $\exists c_1, c_2$, so dass + $c_1 \cdot |g(n)| \le |f(n)| \le |c_2 \cdot g(n)|$. + \section{Graphentheorie} \subsection{Kreis}\label{graph:kreis} Ein geschlossener Kantenzug mit $k_1, k_2, \dots, k_s$, der die Ecken $e_1, e_2,\ldots,e_s=e_0$ miteinander verbindet.